Future Vision of Mathematics

MW: Yes, and in children as well, with their favourite numbers, and feelings about each of the first few positive integers — ethnomathematics and children that one finds in 'primitive' numerological systems.

C: Something that gets beaten out of people by mathematics: when people start learning mathematics, it’s as if the first task is to extirpate any idea that numbers have quality.  Mathematics is in fact often seen as constitutively opposed to any such intuition.

MW: Yes. Marie-Louise von Franz, one of my favourite writers, who studied under Jung and wrote a lot about number archetypes, she talked about number having both quantitative and qualitative aspects.  The quantitative is obvious, we all use numbers to count.  Cultures who revere certain numbers and have mystical beliefs about them which we might laugh at, they still use them to count with and to trade, they recognise that they have a quantitative aspect.  This is the aspect of number that has given rise to economics and technology; but equally, perhaps even more importantly, there’s the qualitative aspect that only survives in our culture in children having favourite numbers, some adults having lucky numbers, not wanting to sleep on the thirteenth floor of an hotel, the way they might choose lottery numbers, that sort of thing.  But, you know, in ‘serious’ society numbers are supposed to be entirely quantitative.  Von Franz wrote about a traditional Chinese story involving eleven generals who, faced with some very difficult military situation, took a vote as to whether they should attack or retreat. Three voted to attack and eight voted to retreat.  So what did they do? They attacked, because three was a more favourable number — it wasn’t a bigger number, but it was a number associated with unanimity, or some other favorable quality like that.  And the attack was a success.

So it’s interesting that they could build a civilisation that was able to have a functioning economy and military and to govern millions of people — clearly they were intensely aware that number had a quantitative aspect — but there was also a serious engagement with the qualitative aspect which is dismissed in our present culture as entirely superstitious.  Now I’m not encouraging people to engage in completely arbitrary numerology, I mean, I’ve looked at a lot of that new age numerology literature, and the problem is, nothing can be verified: someone can write a book saying a particular number means something, and someone else can write another one saying it means the complete opposite.  It just confuses matters, as there’s never any consensus or certainty in these interpretations.  That’s why professional mathematicians would almost unanimously just react against it and say it’s all rubbish.

C: But is there any way to talk about it which doesn’t get into that morass of mysticism?

MW: There are two approaches: one is the serious attempt by Jung and his followers to catalogue all of the ethnomathematical systems, undertaking a serious study and survey of various cultures and their relationship with number, trying to find common threads, and through psychoanalytical work and dream studies, trying to find extract essential patterns to build up a body of material from which we could possibly deduce something about how number interfaces with the psyche at a fundamental level.  The other approach is to seriously study number theory, because as far as I’m concerned, that is numerology, really — you’re looking at the properties of integers, and if you study it to a certain depth it takes you into the realms of what you could only call the mystical or the uncanny, where cracks seem to open up in your normal understanding of reality.

C: Is that perhaps what characterises number theory as opposed to mathematics, what makes it a very different discipline?

MW: Well, number theory is universally acknowledged as a branch of mathematics.  It can’t really be separated from it like that.  But it arguably has a unique status at the very heart of mathematics.  You’re working at the very root of it all, dealing with the simplest objects, the positive integers.  And yet you come across these counterintuitively complicated structures and results.  You can separate mathematics into branches and disciplines but they all ultimately overlap and interrelate. Gauss (who himself was called the ‘prince of mathematicians’) called mathematics ‘the queen of the sciences’ , and number theory ‘the queen of mathematics’.  The idea is that number theory is generally seen as the pinnacle, in that it contains the most difficult problems; also it’s concerned with the integers, and all of the rest of mathematics ultimately relies on integers.  Hence it’s not surprising that problems of number theory do seep into other areas of mathematics, and even physics.  What is surprising is that physics is beginning to shed light on number theoretical structures like the zeta function, as if it were just one of a class of objects, whereas it’s meant to be this fundamental object underlying everything. What I’m trying to describe with my clusters of bubbles isn’t intended as any sort of serious mathematical proposition, it’s just a picturesque visualisation — trying to look at the number system from another angle, if you like.  But there’s a hidden assumption within the Peano axioms, I think, which needs to be addressed — although I don’t think I’m the one to address it.  It concerns the axiom which allows you to always add one.  Even in the proof of the infinitude of primes, I sense some sort of subtle circularity there — the idea is that, if the number of primes were finite, you could multiply them all together and then add one.  And that rapidly leads to a contradiction concerning primeness and divisibility…hence there must be infinitely many primes.  So that takes you back to the Peano axioms, the idea that you can always add one. But in my visualisation, multiplying them all together would correspond to building one mighty cluster using one of each type of bubble.  And in that visualisation ‘adding 1’ is a far less obvious operation.  This ties in with problems of time, the idea of time, repetition, even basic physical questions: you know, this ‘adding 1’ business presupposes that you’ve got a physical space, something like the space we’re familiar with, in which you can make a sequence of marks, or a time continuum in which you can make a sequence of utterances or beats. And I feel there may be subtle assumptions concerning the homogeneity of time and space involved in this, too.

C: These questions of time and space must fall out from the primes’ intimate connection to the relationship between multiplication and addition.

MW: Brian Conrey, who’s President of the American Institute of Mathematics, and Alain Connes have both been quoted as saying that RH is ultimately concerned with the basic intertwining of addition and multiplication. And if we haven’t really got a clue how to prove RH — which we don’t — we’re going to have to own up, we don’t even understand how addition and multiplication interrelate. A more succinct, precise way of describing these two possible constructions of the primes that I have outlined — the conventional ‘just add 1’ approach, and my ‘lexicographical’ approach with its equivalent clusters-of-bubbles visualisation — is given by Grald Tenenbaum, who certainly knows what he’s talking about: Addition and multiplication equip the set of positive natural numbers with the double structure of an Abelian semigroup.  The first [addition] is associated with a total order relation as it is generated by the single number one. So if you’ve got addition and you’ve got this single number 1,  you can generate the postive integers just by adding 1 plus 1, 1 plus 1 plus 1, etc.  If you take 1 as your ‘additive generator’, the universe generated is the set of positive integers. The second [multiplication], reflecting the partial order of divisibility,

This probably isn’t the time to get into the subtle issues of ‘order’ in mathematics — you’ve got ‘total order’ and ‘partial order’: addition relates to total order, where something definitively comes before or after something else; and divisibility relates to partial order, a less distinctive type of order, although I won’t get into the details of that… [Multiplication], reflecting the partial order of divisibility, has an infinite number of generators, the prime numbers. So, now, rather than starting with just the number 1 and combining it with itself in every possible way using addition, we start with this infinite set of primes and then take all possible multiplicative combinations.  Defined since antiquity, this key concept has yet to deliver up all of its secrets, and there are plenty of them.

It has the quality of a square peg in a round hole, this tension between addition and multiplication.  It’s almost like, despite the inarguable perfection of the number system, they don’t really fit together very well, and they generate what I feel is something like friction, and this produces the sprawling mass of definitions, theorems, lemmas and conjectures that we call analytic number theory.  There’s a novel by Apostolos Doxiadis called Uncle Petros and Goldbach’s Conjecture — it’s written as fiction, but he gets some key ideas across through an elderly mathematician character.  This is very well put, I feel: Multiplication is unnatural in the same sense that addition is natural.  It’s a contrived second order concept, no more really than a series of additions of equal elements. So that’s the point, that 3x5, you can see that as 0+3+3+3+3+3 — you start with nothing, zero, and add three five times.  So in a sense you can build multiplication out of addition, whereas it doesn’t work the other way around.  So addition is a first order operation, and multiplication is, as he’s saying, unnatural, in that it’s ‘second order’. The thing that struck me about it when I was dwelling on this for a while was that it has to do with time, it has to do with repetition.  And it also relates to the very deep issues concerning the whole idea of where number comes from and how we define number.  As I hinted earlier, I’ve spent a lot of time thinking about how you could ever have two of anything. You know, there are two people sitting here in this room right now, but that relies on the definition of what a ‘person’ is.  We define the category linguistically, and we think we know what a ‘person’ is, but you can imagine some sort of genetically-engineered mutant that may or may not be a ‘person’ depending on how the definition was formulated, and the definition’s made of words and each word is imprecise, is subject to interpretation.  So any type of category you define is going to have a ‘fuzzy’ boundary, so...although it works quite well for day-to-day affairs, counting things works fairly well, you’ve got fifteen sheep in your paddock.  But you can always contrive some convoluted situation where, maybe it’s fourteen sheep or maybe it’s fifteen — is that odd looking creature really a ‘sheep’ or is it something else?  So, it comes down to issues of language and definition.  We consider chunks of spacetime, we recognise patterns and say, yes, that chunk of spacetime falls into suchand-such a category.   As I said, I started to wonder how you can really have two of anything.   Every entity ultimately distinguishes itself from every other, these categories are not mathematically precise, there’s an arbitrary element involved in deciding whether things get included — “where do you draw the line?” as they say.   And yet these categories are the essence of counting, and if there’s aproblem with applying the concept ‘2’ to our experience then there’s going to be a problem with all of the other positive integers. 

Exceptionally, when you get down to the subatomic level you can have two of something, because each individual electron is absolutely indistinguishable from the others.  So that’s interesting, that this concept makes sense at the subatomic level but then  ‘fuzzes out’ at macroscopic scales.   But the thing is, when you say ‘3x7’, you’re effectively saying ‘three sevens’.  So, seven pebbles in a row — you count out seven by adding one plus one plus one, etc. That feels quite ‘natural’.  But then, to make the leap to ‘three lots of seven’…you can have three giraffes or three potatos, the fuzzy boundaries mean that’s a difficult enough issue as it is, but ‘three sevens’ presupposes that a ‘seven’ is something that there can be more than one of in some sense...

C: One would have to say that the multiplier and the multiplicand are somehow of a different order, two different types of numbers are involved in the operation.

MW: Yes, one is operating on the other.  If you add, it doesn’t matter...I mean, it’s true to say that 3x7 is the same as 7x3, you’ve got this basic ‘commutative’ property applying to the positive integers.  But when you consider the ‘act’ of 3x7, the three is how many times you’re doing something, whether it’s laying out a row of seven beans or playing seven drumbeats, and the seven is some kind of an extension in space or time.  3+7 or 7+3, both numbers play the same rôle.   So there’s something there, not easy to pin down, which we don’t understand, and I have a very deep sense that we won’t really understand it until we really understand time. It has something to do with time.  Our inability to understand the primes, our inability to prove RH is a symptom of our inability to understand the relationship between addition and multiplication, and that is related to our relationship with time.

C: On your site you quote J.J. Sylvester: I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time.

MW: I think he must have been thinking about the relationship of multiplication and addition in terms of time. This was 1888, so RH had been posed, but mathematicians long before RH understood that the enigma of the prime numbers was rooted in the uneasy relationship of addition and multiplication.  So possibly he had a

sense that the relationship had something to do with time. But he says ‘the profound mystery which envelopes our conceptions relative to prime numbers’ — in other words, the puzzling interface of addition and multiplication — ‘depends upon the limitations of our faculties in regard to time’.  So if there were a higher dimensional, a two-dimensional ‘time surface’ or something like it — the word ‘superficially’ is being used by Sylvester in the original sense meaning ‘relating to surfaces’ — our minds, normally constrained to a ‘timeline’, could perhaps ‘spread out across it’ in some sense.  It’s perhaps a bit like being able to come up off the surface of the earth and look down from a third dimension to get a sense of how things are laid out, whereas when you’re stuck on the ground, certain things are not at all apparent...but these are all very vague and intuitive ideas. * C: In one of the papers you link to in the archive11, Volovich suggests a most extreme and startling explanation for the concurrence of physics and mathematics.

MW: Yes, and you may have noticed that he quotes Pythagoras at the beginning, a slightly amusing Greek-to Russian-to-English compound translation of “all is number” — “the whole thing is a number”.  I got very excited when I first found that paper, because he’s suggesting that number theory is the ultimate physical theory.  That came out in 1987 as a preprint at CERN — he’s an accomplished physicist — but it was never published in a journal.  The fact that it never got published and the fact that he hasn’t responded to my questions about it could suggest that he’s backed away from it somewhat.   I can’t speak for him, but I wonder if he’s slightly embarrassed by its more grandiose claims, in the way I was suggesting earlier that physicists and mathematicians can be. But the thing is, he has done this vast body of work on p-adic physics, which I referred to earlier. And the rise of p-adic physics is a very interesting thing in itself because, you see, even though the universe at the scale of this room is Archimedean — I can lay my ruler end to end and will eventually reach the end of the room — the universe is not Archimedean at all scales.  Below the Planck scale, it’s no longer Archimedean.  Below this 1035m or so — which to some people sounds too small to worry about, but you just take a metre, then a tenth, then a tenth, not that many times, really…It’s not that our instruments aren’t precise enough to measure below that scale, it’s that the whole idea of measurement as we’ve formulated it ceases to make consistent sense.  And effectively, space becomes non-Archimedean below that scale. There’s a similar scale with time and other fundamental quantities, below which they become non-Archimedean. You can theoretically join some unit of measurement end-to-end and never achieve a given, finite extension. This has led people to think that maybe p-adic physics, where you’re dealing with a non-Archimedean number system, would be more appropriate for application at the sub-Planck scale.  And Volovich seems to be suggesting that different non-Archimedean number systems could apply to different regions of space and time at different scales.  Again, I’m not entirely sure: large parts of the paper are beyond the scope of my present understanding.  I’m intrigued by his referenced to ‘fluctuating number systems’, but I don’t know whether he means fluctuating with time, or in some other more generalised sense.  People are now starting to think about applying p-adic mathematics to the physical world.  Each p-adic number system provides a different sense of ‘distance’ between two rational numbers, and that notion of distance then allows you to define all the other numbers which aren’t rational via precise mathematical concepts involving ‘limits’.  I mentioned this earlier. This distance or ‘metric’ is defined in terms of divisibility of primes.  It has to do with highest powers: for instance, in a 7-adic metric, finding the distance between two rationals involves basically looking for the highest power of 7 that divides into the numerator of their difference — that difference of course is also a rational number — when it is expressed as a fraction in lowest terms.   As a result of that, number theory comes flooding into your p-adic physics: if you start looking at p-adic or adelic space and time, issues associated with the prime numbers become directly relevant.  Of all of this number theory/physics material I’m archiving this is the area I’m least familiar with.

C: Saying that the means of measurement, that the possibility of measurement has changed is one thing, but saying that numbers are actually the ‘atoms’ themselves, so to speak, is something else: that means that there is no longer some thing you’re measuring.  The measurement itself takes on a sort of substantiality.

MW: Yes, these are very difficult notions to grasp, in so far as I understand what’s being proposed.  I think, perhaps like myself, Volovich caught a glimpse of something, got quite excited about it and wrote it down; he’s quoted Pythagoras — it’s as if there’s some mystical quality to his insight.

C: There might be thousands of these papers hidden everywhere that people haven’t published.

MW: I’m not sure it would be in the thousands, but who knows…There’s a general hesitance to stick one’s neck out.  If I’m helping to encourage that sort of thing, then I suppose that’s a useful contribution.

C: Exeter University has granted you an honorary fellowship and hosts the web- archive, but there is no funding available for your work.   Apart from your own fascination with the subject, what drives you to continue this labour of archiving and making your own speculative connections public?

MW: Over the years after I’d dropped out of formal academia, I spent a lot of time thinking through and honing these ideas about mathematics being some sort of inner priesthood of our scientistic culture that’s in the process of destroying the ecosystem, and wondering what could be done about it, how do we change this, you know? I felt that campaigning to stop the destruction of this or that rainforest isn’t going to be enough, you’ve got to go right to the core, to the root of the problem, the fulcrum.  And, reading von Franz, with her ideas about ethnomathematics, and quantity and quality, and reading René Guenon, who — although I don’t embrace his traditionalist fundamentalism — wrote a fascinating book called The Reign of Quantity and the Signs of the Times, I started having this idea that only when Western Culture re-evaluates its relationship with number can there be any real change in the way we relate to the world, because we’ve got stuck in a ‘quantocentric’ view of the world.  And so I have felt at times that what I was trying to bring forth — whether it was in my strange 1998 ‘evolutionary’ notion or just in my networking of various people’s work via my web archive—was an acceleration towards an imminent transformation in our relationship with the number system.  I was quite driven for a while,  but I’ve become considerably more cautious and sober in my approach to this since.  I saw what I perceived to be clues...felt that it had to be coming, and only through that sort of transformation will the Western project ever be able to steer itself in a less destructive direction.  At times I’ve felt that I had an important rôle to play — not that I was ‘chosen’ to do it or anything, but that my work was cut out for me, and it was an important mission.  Other times, I’ve been much less certain, and wondered, you know, why am I sitting in front of this computer editing HTML, when I could be spending the same time and effort campaigning for, say,  the rights of an indigenous tribe having its land ravaged by a multinational corporation.  I had to justify this to myself when people I knew were involved in things like that, by telling myself, well actually they’re just dealing with the symptoms, whereas I’m trying to deal with the root of the problem.  So it verged on an idealism, almost an activism. 

C: The point being that rather than lamenting the destructive rôle of number and of science, one tries to recognise that there’s something else within number, and as you said, to re-evaluate our relationship to it, which is not to say to reject it, but to become more numerate...

MW: Yeah, which is what I saw around me, people being very suspicious towards mathematics, hating it, seeing it as controlling and evil, and I thought, no, we need to get inside it, try to understand where it comes from and how it works.  But then I started to question whether I was just  creating a whole set of complex and noble motivations for myself when in fact it was just my ego or desire to be acknowledged for what I’d achieved, or, you know, just wanting some sort of recognition or status.  I was continually wondering what it was that was motivating me, and trying to rein myself in and consider the worst possible motivations as well as the best. I had a kind of motivational collapse in early 2005, when I was struck by a very deep sense of there being insufficient time; you know,  I had this grandiose hope of helping to effect some sort of long term change in culture and the way in which we deal with the number system. I started to think, maybe what I’m contributing to would have that effect if there were a few more centuries left of relatively leisurely culture and well-funded academia to take these ideas on and develop them, but, you know, we’re facing multiple global crises, and this sort of thing is never really going to have time to take root.  I’ve since drifted in and out of this activity periodically, found what I think is a healthy level of interest in these matters.  But I don’t strongly believe that I’m part of some current of cultural change anymore, I’m just...I suppose you just can’t know what effect you’re having, particularly with the Web, when you’re pushing ideas out. You don’t know who’s reading them and what they’re going to do with them — a bright teenager who reads my website might be inspired to study mathematics and, influenced by some of the hints, clues, suggestions, etc. I’ve assembled, go on to make amazing discoveries...who knows? There’s also the whole relationship between psyche and matter which seems to have been at the centre of all my interests over the years. I got involved in parapsychology for a while, online psychokinesis research in 1996, wondering whether there really was something in that, and what it would imply concerning the psyche-matter interface.  There’s also a very exciting interdisciplinary field of ‘consciousness studies’ emerging, and which I’ve been following, people trying to understand the physics of consciousness, looking at microtubules in brain cells and how quantum mechanical phenomena at that scale might help to explain the origins of consciousness — physicists, neurologists, philosophers, psychologists, anthropologists, psychopharmacologists, etc. are all contributing to this field.  Then there’s all the Jungian  theory concerning myth, archetype, synchronicity and the ‘psychoid’ level of reality — a kind of psycho-physical interface.  The simple fact that mathematics is able to describe the world at all, that’s a mystery involving mental constructs being mapped mapping onto material reality.  There’s the ‘mind-brain problem’ which philosophers debate.  And then dreaming, shamanism, schizophrenia, quantum-mechanical paradoxes, these are all things I’ve spent a lot of time thinking about, reading about — generally wondering how it all fits together.  And it had occurred to me that these topic cluster around the central mystery of how matter and psyche interface. But I’d been thinking about prime numbers, etc. for a few years before it occurred to me that this is very much part of the same picture.  I’d been exploring the interface of physics — which concerns matter, obviously — and number theory, which, as that Tenenbaum quote suggested, is really an exploration of  ‘the mind itself’.   And the research I’ve been interested in archiving displays a two-way traffic: Number theorists have been providing concepts and structures which physicists have used to better understand the world of matter.  Physicists have been able to, using their understanding of matter, shed light on the internal workings of the number system. Even number theory without the physics is implicated: although number is widely considered as a mental construct, at the same time it manifests directly in the world of matter: when you consider a quartz crystal or a five-petalled wildflower, it’s hard to deny there’s an essential ‘sixness’ or ‘fiveness’ there.  So, number itself is a bridge of sorts between psyche and matter. This last idea, that number is a bridge between psyche and matter, comes quite close to something Jung was exploring in his later career.  He left a lot of incomplete work when he died, and I believe he left von Franz to look at number archetypes.  He’d looked at individual integers, the first few integers and their various associations.  But later, more importantly, he’d come up with the idea that, not individual numbers with their associations, but the set of positive integers as a single entity is in itself an archetype, the archetype of order.  Now what has distinguished Western culture from the rest of humanity, what characterises the Sumerian-toBabylonian-to-Greek-to-Roman-to-Western-European cultural current that dominates the planet with its measurement and science and so on, is the way we’ve dealt with this archetype which normally inhabits the collective unconscious. I picture it as a sort of mysterious sea creature — we’ve hooked it and we’ve hauled it out from the dark depths into the daylight of consciousness. We’ve taken something that was primarily unconscious, and which would naturally manifest primarily via the number archetypes and number associations in other cultures. We’ve dragged this thing out of the sea and onto the land, cut it up and studied it, studied its anatomy in great detail in order to obtain a new kind of magic, if you like, and that, I came to believe, was the root of all the world’s problems.  But then we have this emergence into consciousness of the set of the prime numbers buried within the set of positive integers, a hidden archetype within an archetype, a kind of chaos within order, the black dot in the yin half of the yin-yang symbol; the emergence of that archetype — the prime numbers, the zeta function and everything they entail — into mass consciousness, is just starting now, really.   The first four ‘popular’ books on RH have all come out in the last couple of years...it’s strange that this should all be happening so suddenly.  Thinking along the quasi-Jungian lines I’ve sketched out, the integration of these ideas into consciousness, the idea of the Riemann zeros having their origins in some ‘older’ or ‘deeper’ numerical reality, something more ‘primordial’, etc. may turn out to be of profound historical significance. According to the insanely optimistic wishful thinking which I’ve since distanced myself from, this could be the event that would start to alleviate the effects of rampant ‘quantocentrism’ and put things back into balance.

C: I wonder whether the growth of ‘popular science’ could play a rôle here — thinking in particular of the many books which have been published on RH.

MW: The fact that you’ve got four books on RH out suddenly — why is this, why hadn’t this happened before?  I’m sure a few years ago most people involved would have said that it’s impossible to explain RH to laypeople.  But four authors have done their best, with varying degrees of success.  The books have all been well received, have sold fairly well.  So why is this happening? The mystically inclined might invoke an unseen force that’s trying to bring these ideas into consciousness. Jungians might talk about ‘compensation’ and the collective unconscious. But more simplistically, more materialistically, it’s market forces, it’s capitalism, and it’s because people are looking for meaning.  Many are turning to New Age cultism, some are turning to born-again Christianity, Scientology, fundamentalist Islam, whatever. But there are a lot of people who are aware that the real ‘guardians of truth’ these days are not priests and monks, but scientists and mathematicians, and yet, they find themselves in a position where they don’t know anything about the essential subject matter.  So they want someone to explain, say, the mysteries of quantum physics to them. I get this all the time, people really wanting me to explain quantum physics, fractals, relativity, the golden mean, chaos theory, p; there’s a handful of things that people get really excited and obsessed about, you know.  And of course the market system rises to meet a demand, a growing demand for meaning.  The problem is that capitalism doesn’t care whether a book is accurate or well-written, it just cares about sales figures.  So as a result you get gross oversimplifications hitting the market and sometimes selling quite well.  Because the market has expanded, there is more competition, and ideally, if you believe in the effectiveness of capitalism, then the ‘best’ stuff will float to the top — but ‘best’ in this sense doesn’t necessarily correlate with truthfulness or accuracy, rather with how successfully the book quenches readers’ thirst for meaning.  There does seem to have been a certain amount of progress, though.   I don’t really watch much TV, but it does now appear that with the computer graphics available, it’s possible to make some things a lot more visually accessible, so viewers can at least get a flavour of the problem, or of what’s at stake.  But the really deep stuff, the major philosophical problems underlying maths and physics…it’s hard to imagine that there really is a shortcut to years and years of disciplined study. I mean, you might be able to get the basics of something across to a few, a small section of the population who are already interested and whose minds are structured in a certain way — it’s not to do with levels of intelligence, just a certain kind of intelligence.   You’ve got committees for the popular understanding of science and things of that nature, but they’re very marginal. Unless there were a major cultural shift, unless you had major government funding, and the top layer of mathematicians and scientists committing themselves full-time to bringing this stuff through into popular culture...but there’s no motivation for that to happen — governments aren’t interested in educating their populations except in ways which will further economic growth.  They want a certain proportion of young people to be trained up to be economists, accountants, engineers, etc. ‘Truth’ doesn’t really come into it.  So I doubt it…but, again, you never know, some major cultural shift could occur where the demand for this sort of knowledge reaches the point where the best people would feel obliged to provide it. Or, possibly, there could be some sci-fi type breakthrough involving direct brain-to-brain knowledge transfers, you know, you can’t rule these things out, but I’m not holding my breath! You’ve probably noticed, part of my website is very formal-academic, the web-archive aspect; and part of it is just about getting fundamental ideas across to people who are open to them and just want to understand their reality a bit better.  I have felt in the past, with my ‘activist’ hat on, that it’s important to bring some of these issues to widespread public attention — the basic issues of the number system.  At this stage I don’t know if it is ‘important’ or not, but I’d be very interested to know what the overall effect of that kind of exposure would be. Again, I suppose I am still gripped by the idea that, if we transform humanity’s relation with number, that could have a positive transformative effect.  I suspect I’m still partially motivated by that belief at an almost subconscious level. The only thing I can really say with any confidence at all is that I think we’re on the verge — and again, the timescale is very indefinite here — but Western Civilisation is on the verge of collectively realising that the number system is something very different from what it had previously thought it to be.  I haven’t got a

particular theory about what it is, I just know it isn’t what we think it is.

PC I applaud the efforts that you have made to draw attention to many existing problems that relate in various ways to our “quantocentric” culture.

And though interest in this number related field because of its unique intellectual requirements remains a small niche area, I believe that you have been more successful than you imagine in becoming an influential figure in this regard.

Your “Number Theory and Physics Archive” and the “Secrets of Creation” trilogy are an excellent contribution to the field. Though there may never be a mass readership for the books in the trilogy, they can still exercise an important influence on the limited number of people who can recognise the supreme importance of the topic.

And I am delighted to see that your  “Prime Evolution” is now available again as it is a very fine contribution. I remember accessing it some years ago on the web and then it seemed to vanish only to reappear again more recently. So I hope many more people will come to discover this superb discussion. 

In terms of the damaging effects present understanding of the number system is having on our culture we would both share a great deal of common ground. I do however hold a much stronger conviction that mathematics is itself in need of radical reform. And this relates to the urgent requirement to discover its unrecognised holistic aspect, which apart from potentially opening up a vast new qualitative field of enquiry has profound implications for the interpretation of existing quantitative relationships. 

This crucial holistic aspect has remained greatly repressed throughout its history and especially with respect to the increasingly specialised development of mathematics in the past two centuries. From a psychological perspective, it therefore entails that the hidden unconscious aspect of understanding, which affects every single mathematical issue must now be slowly brought into the conscious light, before then becoming properly integrated with its conscious counterpart.

This will entail the most radical revolution yet in our intellectual history, which I believe will coincide with a great new spiritual awakening. However apart from a few lone voices crying in the wilderness, it will only properly emerge after a protracted series of crises that gradually awaken the world to the great existing lack of a true holistic appreciation of reality.

So I will conclude these comments now with some personal reflections on a future “golden age” of Mathematics.

Here, I would see three distinct domains (where only one currently exists).

1) The first - which for convenience - can be referred to as Type 1 Mathematics, relates to the traditional analytic approach based on the reduced quantitative interpretation of mathematical relationships.

At present, in terms of formal recognition, mathematics is exclusively identified with this type.

As its methods have become increasingly specialised in an abstract rational fashion, admittedly enormous progress has been made. And this will continue into the long distance future with many new significant analytic findings for example regarding the Riemann zeta function (and associated L-functions).

However an important present limitation, as we have seen, is the manner in which exclusive identification with the analytic, blots out the holistic aspect (with which creative intuition is more directly associated).

While not wishing to prevent further progress in analytic type developments, eventually I believe it will be accepted that the Type 1 represents just one highly important aspect, which should not be exclusively identified as mathematics.

So a strictly relative - rather than absolute - interpretation will thereby eventually emerge for all its relationships.

2) The second - which I customarily refer to as Holistic Mathematics - represents the Type 2 approach.

There have been some precursors in this regard notably the Pythagoreans. However beyond a number of suggestive ideas, they never really succeeded in the explicit development of the holistic aspect, which represents an urgent requirement now for our present age. 

My personal development has been somewhat unusual in this regard. Though displaying a marked ability for mathematics as a child, serious reservations with the standard treatment of multiplication arose at a very early age. Therefore, from that moment I was already reaching out for a new holistic dimension, not catered for in formal terms.

So mathematics for me has very much represented a solo voyage of unique personal discovery.

And in adult life, I have largely concentrated on elaborating the hidden holistic dimension of mathematics, which is truly enormous in scope. Then in the last 15 years or so, using these holistic insights, I have turned my attention to topics such as the Riemann Hypothesis with a view to providing a radical new perspective as to their inherent nature.

Though retired from work as a college lecturer in Dublin for nearly six years, it feels only now that I have come full circle in being able to finally resolve (at least to my own satisfaction) those childhood questions regarding multiplication.

However it is very difficult indeed attempting to convey holistic mathematical insights to a professional audience that does not formally recognise their existence.

So my most successful communication has been with talented generalists from diverse mathematical backgrounds, seeking to understand various intellectual disciplines in a more integrated manner.

However even here, there has been considerable resistance to the belief - reflecting the deep-rooted nature of conventional assumptions - that mathematics itself is in need of radical reform.

Bearing the above comments in mind, I will now try to convey some of the flavour of my holistic mathematical pursuits.

As it is directly concerned with the qualitative nature of symbols, much of this work has related to the clarification of the various stages of psychological development through the holistic use of these symbols.

In terms of the physical spectrum, natural light forms just one small band on the overall spectrum of energy.

In like manner, the mental structures (based on accepted common sense intuitions underpinning linear logic) represent just one small band on the overall potential spectrum of psychological development.

Conventional mathematics represents specialised understanding based on this linear band.

However just as there are further forms of physical energy (besides natural light) equally there are further forms of psychological energy (besides the accepted intuitions of conventional mathematics).

These further bands on the psychological spectrum have been extensively investigated by the major esoteric religious traditions East and West, where they are identified with advancing levels of spiritual contemplation of an increasingly formless nature.

However what has not yet been realised - except in the most perfunctory manner - is the important fact that these bands are likewise associated with new forms of holistic and radial mathematical interpretation (utterly distinct from conventional type notions).

In analytic terms, the study of higher mathematical dimensions entails rational understanding of an increasingly abstract nature (where the object aspect is increasingly separated from its subjective counterpart).

However in holistic terms, the study of higher dimensions by contrast entails appreciation of an increasingly refined intuitive nature, indirectly transmitted in paradoxical rational terms, where both objective and subjective aspects are seen as ever more interdependent with each other.

Now one of the extraordinary findings arising from these investigations is that all psychological and physical structures, which are of a dynamic complementary nature, can be given a distinctive holistic mathematical rationale.

And the holistic notion of number is intimately tied to these structures.

So from the holistic perspective, accepted formal mathematical understanding is 1-dimensional (in qualitative terms). This simply means that the qualitative aspect is formally reduced in a quantitative manner.

However as I have already mentioned, associated with every other number (≠ 1) is a distinctive means of interpreting mathematical symbols with a partial relative validity. So the absolute type understanding that we accept as synonymous with valid mathematical interpretation represents just one special limiting case of a potentially infinite set.

And this insight was later to prove of inestimable value in relation to a true dynamic appreciation of the Riemann zeta function.

Also, the holistic relative notion of number is intimately connected to a corresponding new holistic appreciation of the nature of space and time with a direct relevance in physical and psychological terms.

This has immense implications for physics where space and time notions have been reduced in a distorted manner corresponding to a merely quantitative interpretation of number.

So we apply general impersonal notions of space and time in scientific terms. Not surprisingly, physical reality then appears to us as lacking any personal aspect.

However properly understood, all reality contains both personal and impersonal aspects. So the qualitative personal features of nature relate in holistic manner to unique configurations of space and time (which find no place in the conventional approach).

Indeed one could validly argue that the root of the present ecological crisis, which is so rapidly unfolding on our planet, lies in the merely quantitative appreciation of scientific reality (which has greatly distorted its true nature).

For example, as we have seen, the holistic notion of “4” relates to a dynamic appreciation of the corresponding 4-dimensional nature of space and time (with positive and negative directions in real and imaginary terms).

In my own work I have come to appreciate how important the holistic understanding associated with 2, 4 and 8 dimensions respectively are for an integral appreciation of reality.

And once again there are strong links with Jung, who came to realise the significance of certain pictorial diagrams termed mandalas – widely employed in Eastern religions as meditative tools – for psychic integration.

The most common mandalas generally entail highly ornate images based on the division of a circle with four or alternatively eight equidistant points.

From a mathematical perspective, these therefore serve as geometrical representations of the 4-dimensional and 8-dimensional holistic interpretations of reality.

However, just as there is a key distinction with respect to the behaviour of the Riemann zeta function for even and odd values of s, likewise there is a key distinction in holistic terms as between the even and odd dimensional values. Basically, the even values relate in more passive terms with the achievement of a certain attainment of integration with respect to experience (entailed by the corresponding dimensional number). The odd values then relate to a more active state of asymmetrical involvement, associated with a degree of linear understanding (which leads to a departure from the previous equilibrium state).

So in ideal terms, human development should continue moving in and out of progressively more advanced stages of psychic integration.    

In brief, akin to directions on a compass, all the various holistic dimensions can then be looked on as providing unique configurations of the dynamic relationship as between wholes and parts (in objective and subjective terms).

Just one final example I will offer, though all this represents but the tiniest glimpse into a potentially vast new field of investigation, is the holistic counterpart to the accepted binary system!

So the two binary digits 1 and 0 - given an independent interpretation in the standard analytic manner - can be potentially used to encode all information processes.

However, equally the two binary digits 1 and 0 - now given a holistic interdependent meaning as linear (1) and circular (0) type understanding respectively - can be likewise used potentially to encode all transformation processes.

So for example, in this contribution, I have argued that the number system - and indeed all mathematics - should be interpreted as representing a dynamic interactive transformation process, entailing both quantitative and qualitative aspects.

And these two aspects relate directly to 1 and 0 respectively (in a holistic manner).

In fact, at the most general level all differentiated and integrated processes both in physical nature and psychologically in human terms are encoded in a holistic binary digital manner.

We are now living in the digital age where so much information is already converted in a binary fashion (entailing the analytic interpretation of 1 and 0).

However what greatly concerns me is that because of the dominance of mere quantitative analytic type understanding with respect to such technology, that no adequate means for corresponding transformation exists in our society (which requires the corresponding development of holistic type appreciation).

So at present, a number of related crises are developing in environmental, economic, political and social terms all of which have at their root an inability to tackle global issues in a truly integrated fashion (which demands a radical holistic approach).

And this in turn is deeply related to a distorted view of science which unfortunately is treated as the new world religion. However, at an even deeper level (though not properly recognised) it is due to a distorted view of mathematics which underpins the scientific approach.

3) This, which I commonly refer to as radial mathematics or more simply the Type 3 approach, represents potentially the most comprehensive form of mathematical understanding, entailing the coherent integration of both analytic (Type 1) and holistic (Type 2) aspects.

When appropriately understood, all mathematics is intrinsically of a Type 3 nature, though not yet recognised because of the deep-rooted acceptance of reduced assumptions.

Indeed it is only in the context of radial mathematics (Type 3) that the other two aspects, conventional mathematics (Type 1) and holistic mathematics (Type 2) can reach their fullest expression. So perhaps some day in the distant future, Type 3 will become synonymous with all mathematics.

However even within this category, I would distinguish three important sub-types, (a), (b) and (c) respectively.

Though rooted to a certain degree in the holistic aspect of mathematics, sub-type (a) is mainly geared to the derivation of exciting new analytic type discoveries (with creative insight playing a key role).

There is no doubt that implicitly, Ramanujan represents a truly extraordinary example of such holistically inspired mathematical discovery. At a deeper level, I believe Riemann also perhaps belongs, leading to highly original discoveries that relied on a strong intuitive dimension. However, unfortunately, in neither case was the holistic dimension of mathematics explicitly recognised.

So in the future, even greater creative analytic discoveries in various fields will be possible, when mathematical talent is backed up with a truly mature holistic appreciation of symbols.

I should perhaps stress that with respect to mathematical understanding that analytic and holistic type abilities are utterly distinct. Thus for example a mathematician with an obvious recognised mathematical talent in analytic terms, may have little capacity for corresponding holistic appreciation.

Likewise it works in reverse so that a person who in conventional terms may display little or no mathematical ability could in fact be potentially gifted at the holistic level. Jung for example would readily fall into this category.

However, it would not be possible for someone to successfully operate with respect to Type 3 mathematics without being able to combine in varying degrees both the Type 1 (analytic) and Type 2 (holistic) aspects.  

The second sub-type - while requiring appropriate analytic appreciation (the degree of which depends on the precise context of investigation) - is mainly geared towards the holistic interpretation of mathematical objects.

I would classify my own recent efforts as a most preliminary version of sub-type (b), operating necessarily at a very rudimentary level.

However this is still adequate, for example, to outline the bones of a distinctive dynamic appreciation of the number system. This then provides the basis to radically re-interpret the nature of the Riemann zeta function (with accompanying zeta zeros and Riemann Hypothesis).

Thus, subtype (b) is not geared directly to analytic discovery, but rather a coherent dynamic interpretation of mathematical relationships. However because it is most creative at a deep level of enquiry, indirectly it can facilitate exciting new directions for analytic discoveries.

The final subtype (c) entails the most balanced version of both (a) and (b), opening up possibilities for the finest form of mathematical understanding, that is at once immensely productive and highly creative and readily capable for example of synthesising various fields of mathematical study.

One of its great benefits is that it can also provide the important capacity to appreciate the potential practical applications of new mathematical discoveries.

The reason now why so much abstract mathematical analysis seems irrelevant in practical terms is precisely because it is understood in a manner that completely lacks a holistic dimension.

However, with both aspects (analytic and holistic) properly recognised, the practical applicability of abstract mathematical findings would be more readily intuited.

And from an enhanced dynamic perspective, all mathematical findings which, when properly understood, are experientially based, have practical applications!

In this regard as a general principle, I would imagine that what is considered most important in abstract terms, is likewise potentially of greatest significance both from a holistic and also applied mathematical perspective.

Another great advantage of subtype (c) is that by its very nature, mathematics can now become readily integrated with the rest of human experience, allowing for the fullest expression of personality development.

Thus if we want a vision of what mathematics might look like at its very best, we would choose subtype (c) with respect to the Type 3 aspect.

However, we are still a very long way indeed from realising this wonderful reality, with the great lack yet of an established holistic dimension to mathematics, serving as the chief impediment.

Other Important Issues

C: This is a striking aspect of your research – the presentation of it is very open: no need to hold back until you’ve got an completely solid hypothesis and then put it online tentatively as a preprint.  The site is continually updated, and you’re creating this network which connects together all these scientists who it seems are working on related problems but don’t always know of each other: in some cases you’re actually notifying them of each other’s work.

MW: I’ve spent a lot of time emailing relevant researchers and alerting them to the existence of new articles or preprints which they may well be interested in. And it’s difficult to quantify, but I do seem to have stimulated a certain amount of interdisciplinary work. I’ve created a rôle for myself which hasn’t really got a name yet, and as far as I know, no-one’s prepared to fund me, but I’m doing my bit to weave together these threads of research. 

Part of what caused my disillusionment with mathematics, which caused me to drop out in the first place, was...well, the overriding impression was the biblical image of the Tower of Babel.  It occurred to me that if you were to put the names of all professional research mathematicians in the world into a hat and pick out two, the chance of there being any real overlap in their research interests would be quite small, and this continues to get smaller.  It was as if mathematical research was getting so fragmented that there was no longer any effective communication possible.  So in a way, I suppose what’s needed, if one wants to try and fix this, is people who are not specialising, but rather trying to get an overall picture and to weave it all together by creating lines of communication.  I didn’t come into this with that intention, but that seems to have been the rôle I’ve created for myself.  I haven’t got any answers at all. I just feel that there are questions that are important and which aren’t being asked – possibly because there just isn’t the language in which to ask them coherently yet. But at the same time, because there are no real constraints on me, I don’t have to prove myself to anyone, publish anything, or stay within any particular boundaries, I can just throw out certain ideas, get people thinking about things, suggest connections between things in such a way as to indicate the existence of something which we can’t yet pin down perhaps, but which will come into focus the more we look at it.  In the mathematical community, at least the proportionally small number of people I’ve communicated with, I do get a sense that there’s a sense of wonder there which is something unquantifiable, something that you couldn’t prove a theorem about, but which is nonetheless there.  It’s something to do with these individuals’ emotional, psychological or even spiritual orientations, I suppose.  But a lot of mathematicians, I’m afraid, do tend towards the familiar stereotype of socially inept, almost mildly autistic people who have very little time for the unquantifiable aspects of life.  And so there is an almost scathing disregard from some quarters. I think – I feel – that anything that’s vague or a little bit ephemeral, they see that as worse than useless, perhaps because their own self-esteem and status is tied up in a self-image of being the guardians of some sort of absolute inarguable exactitude and truth.

C: Your guiding thread is a fascination with how mathematics relates to reality, rather than with mathematics per se. This seems to be related to the fundamental problematic which appears right at the very origin, you could say the co-origin, of mathematics, philosophy and natural science: with the Pythagoreans, who realised that operations carried out on numbers applied – rigorously, but for them somewhat magically – to natural phenomena, and so put forward the idea that reality was actually nothing but numbers, reality was structured by number. In a sense they put forward a type of mathematical empiricism, i.e. the idea that you could go out and explore the world, and what you would expect to find was relationships between numbers, and you could understand the natural world like that.  Now this came to a catastrophic end with the discovery of irrational numbers...

MW: Yeah, the legendary drowning at sea of Hippasus of Metapontum – it’s fascinating stuff, a pivotal event in human history...

C: Certain aspects of the natural world were shown to exceed number – or number as it was conceived then. Certain quantities which can be mathematically described (the diagonal of a square with side length 1, the area of a circle with radius 1, the golden ratio) cannot be expressed as ratios of integers, they are ‘alogos’ or, as we now say, irrational. After a long period under the influence of Aristotle’s instrumentalism, for which every sublunary physical phenomena was subject to an inevitable degradation, meaning that exact mathematics was applicable only to astronomy, the celestial and sublunary worlds were (blasphemously) reunified, most of all by Kepler, under a single mathematical physics, reinvigorating the Pythagorean dream of a mathematical natural science.  Then in the nineteenth-century mathematics seemed to exceed its reference to the real world, to claim its own autonomous consistency, and any necessary link with the natural sciences was removed, mathematics asserted its independence from any application; its applicability to the physical world even seemed to become a sort of mathematical ghetto. Now, in the work you’re looking at, it seems that we return once again to a Pythagoreanism but with a strange twist...

MW: Yes, something’s been turned on its head.  I’ve been fascinated by Pythagoras and the Pythagoreans for a long time.  Sometimes I think, you know, in a way I’m acting a bit like a ‘neo-Pythagorean’…but as you say, there’s a strange twist there.  I think a lot of people forget, when Pythagoras is discussed as ‘the first mathematician’ , that he had one foot in mathematics and another one in a sort of shamanic, mystical-type reality. 

C: Whereas the Pythagoreans discovered in numbers the semi-divine property of rigorously elucidating nature, we have this experimentally and theoretically-vindicated body of method and knowledge taken from natural science, with whose aid we’re trying to illuminate what now seems like a somewhat opaque and mysterious numerical realm; and there are these things within number which still don’t really make sense. Mathematicians such as Chaitin [see article in the current volume—ed.] have said that mathematics must now become a quasi-empirical practice – this is in relation to his own work, but it might perhaps equally be applied here.

MW: Some of the quotes I have on the site agree: Martin Gardner said something about how some problems of number theory might be undecidable and might need a sort of mathematical ‘Uncertainty Principle’.  Timothy Gowers wrote that the primes somehow feel like experimental data, but at the same time he’s well aware that they are rigidly determined.  We find ourselves in a situation where Michael Berry, studying spectra of quantum mechanical systems, can take techniques he’s developed to classify or better understand certain types of physical systems and apply them to the Riemann zeros, in order to produce a hypothesis that we will get a particular ‘number variance’ in the far reaches of the spectrum of Riemann zeros – then years later, you know, computer power reaches the point where zeros can be calculated at that scale, the ‘number variance’ computed...and the graphs match up perfectly. It’s the first time I’m aware of when a physicist was able to tell pure mathematicians something new based entirely on his familiarity with physical systems. 

C: Does the field then become de facto an experimental one?  You have the a hypothetical physical system which will produce the system of vibrations which the Riemann zeros seem to correspond to.  And the only way to find out whether there’s really any system which is adequate to that would be by experimentation – in the same way that the Higgs Boson hypothesised to glue together the results of quantum physics must now be sought experimentally – hence the construction of CERN’s much-anticipated Large Hadron Collider.  Does someone have to build the Riemann dynamical system?

MW: Michael Berry has said he’s absolutely convinced that, if such a thing is physically possible, someone will make one of these things in a lab, and then the Riemann zeros will actually come out on the instrument readings. But at the moment there’s no-one actually conducting any experiments which are getting anywhere near that, or even attempting to.  You do have physicists taking certain ideas – largely mathematical models intended for physical systems – and applying them to aspects of the zeta function.  There  is an experimental branch of study of course, you’ve got people looking at the Riemann zeros themselves, which contain a wealth of data – we’ve got, I believe, hundreds of billions of them calculated now – this is being done with grid computing8. The gaps between them and all kinds of other things you can measure when you’ve got a set of seemingly random real numbers, are being analysed using a variety of statistical methods, random matrix theory is being applied.  So these are, to some extent, experimental studies.  Marek Wolf (Institute of Theoretical Physics, Wroclaw) experimentally detected a widespread physical phenomenon called ‘1/f noise’ in the distribution of prime numbers. 

The prime numbers continue out to infinity, we’ve known they go on forever since Euclid, but we can only calculate them up to a point.  We tend to think our current computers are ‘powerful’ , and we think we can find ‘big’ prime numbers – you know, now and again one will even make it into the news.  But there’s no such thing as a ‘big’ number, this is what I always try to get across to laypeople – because the number system goes on forever, however far we look, proportionally it’s still an infinitesimal step into an infinite unknown .

C: And, of course, in consequence, no matter how many zeros are found, one never comes any closer to a proof of RH.

MW: Yes, exactly.  There’s the duality between Riemann zeros and primes, and so the same idea applies with the zeros.  We can never calculate more than an infinitesimal proportion of them.  Sometimes I use the analogy of large telescopes: you’re looking out into space, and the more you can see, the more you can deduce about the nature of the universe you livein.  Analogously, we can ‘see into’ the number line a certain distance, what we think is a ‘long way’ – but again, it’s meaningless, really, to say a ‘long way’ or a ‘big number’. Of course we can see further than we’ve ever seen before, so we can detect certain apparent patterns which can give rise to hypotheses that we can then attempt to prove.  Similarly we can look further than ever up the critical line now, and with hundreds of billions of Riemann zeros we can test certain hypotheses and generate new ones.  So there’s an experimental element in that.  But as far as the hypothesised Riemann dynamics goes, the quest to try and pin down something like a Riemann dynamics isn’t really being furthered by experimental science as such, rather the progress seems to be coming from mathematicians like Connes, Lapidus and Christopher Deninger (University of Münster).  But these people – well, certainly Connes and Lapidus – do have a very broad interest in large areas of both mathematics and physics, which is what makes their work so interesting. It would be misleading to suggest that mathematics has become an empirical science, since exact formulations are still possible – even in these more hazy areas – at least we can’t rule out the possibility of exact formulations. But an empirical approach has become potentially useful. In connection with this, I should mention the emergence of probabilistic number theory, which in itself raises huge questions.  Probabilistic number theory effectively started in 1940 with the Erdös-Kac theorem which I mentioned earlier, the discovery that the number of prime factors in ‘large’ integers has a kind of random distribution which follows the Gaussian distribution or bell curve.   That discovery led to a whole outpouring of theorems and conjectures which have collectively become known as probabilistic number theory, where you apply the methods of probability theory, and make use of the key idea that divisibility by a prime p and divisibility by a different prime q are ‘statistically independent events’, one has absolutely no bearing on the other.  When you deal with probability you deal with this idea of independent events – well, these are arguably the most independent ‘events’ there can ever be. Physical events in any well-prepared experiment, you might think they are independent; but ultimately every particle of the universe is gravitationally pulling on every other particle, everything is linked, although the effects are generally negligible and impossible to quantify.  The only place where things are totally independent is in the number system – the divisibility of an integer by two different prime numbers. So here is a place where you can apply probability theory, where everything is entirely exact, where you can let your n tend to infinity and that actually refers to something.  Probabilistic number theory allows you to prove things about prime numbers and about the number system generally, using the techniques of probability theory, and that seems highly counterintuitive. The fact that it works at all raises questions which are more like ‘mysteries’ than formal mathematical problems.  There are three separate areas worth mentioning here: the emergence of probabilistic number theory, the effectiveness of the analogy with statistical mechanics – partition functions, etc. which I described earlier – and then the rôle of random matrix theory, which was developed for modelling subatomic phenomena, but then was accidentally found in the 70’s to apply directly to the theory of the Riemann zeros.  So you’ve got three separate areas of randomness-based thinking, stochastic disciplines if you like.  They deal with large systems which have too many components to keep track of individually – these components must be treated almost sociologically, as populations, and subjected to probabilistic or statistical thinking.  All three areas have been effective in furthering our understanding of the number system.  Now, again, mathematicians would tend to focus on at most one of these things, see what could be achieved and perhaps make a few sober remarks on what it all might mean.   But to me, the fact that you’ve got these three areas, all of a stochastic nature, shedding light on the primes and the Riemann zeros, points to something very strange.  We’ve got primes, the most basic things in the  universe as we experience it – the sequence of prime numbers is the most basic non-trivial information there is, it’s the one thing you can’t argue with anyone about, it’s the one thing all life forms in the universe could potentially relate to.  And yet in some ways they seem to be best understood using a type of analysis more appropriate to weather systems, roulette wheels, boxes of gas, etc. I’ve always thought of probability theory as a slightly ‘tainted’ branch of mathematics for three reasons: Firstly, it’s origins are not entirely honourable – I seem to recall that it has its roots in an historical accumulation of gambling techniques which got distilled into a formal theory by Pascal.  Secondly, it deals with ‘events’, repeatable

 ‘events’, which are categories of physical phenomena, ‘occurrences’ of one type or another which can be quantified, measured, counted, numerically analysed, etc. whereas truly ‘pure’ mathematics doesn’t rely on anything in physical reality in quite this way. Finally, by its very nature, probability theory tends to deliver imprecise information – there’s always a margin of error. And yet this system of thought, which has been developed in order to deal in an approximate way with large, complicated physical systems, seems so perfectly applicable to something which is so fundamental, which is characterised by an absolute precision, and which underlies everything else – the distribution of primes!  It’s as if we’ve got something back-to-front.  It’s similarly interesting that probability should have such a fundamental rôle in quantum mechanics: an ultrasimplified account of what QM tells us is that, insofar as it can be understood as being made of particles, the universe can also be understood as being made of ‘fields of probability’ .  Probability theory in a casino, yes; or in a meterology lab... But prime numbers? The fundamental level of matter?  These are things we instinctively feel should be totally deterministic and rigid.  And to me, this suggests we’re looking at something the wrong way ‘round – something’s been turned on its head.  It’s as if ‘randomness’, or some essential, almost esoteric quality associated with randomness – that quality evidenced in our failure to really understand what we mean by ‘randomness’ – is emanating up from these fundamental

realms.  We’ve been dragging it down from the macroscopic scale, the casino scale, down to this micro-level, in a numerical and physical sense, and finding that it helps us understand something.  But I feel something’s back-to front there.  A mathematician called R.C. Vaughan states in one of my archived quotations that it’s obvious that the prime numbers are random, but we don’t know what randomness is.  And there is a real problem with defining randomness.  There are several definitions, information theorists, probability theorists, have put forward definitions of what it means for something to be random. The definitions overlap to a large extent, but ultimately, when is a string of digits random?  If I give you a block of a thousand 0’s and 1’s, it might look completely random, it might even pass numerous tests run on it for randomness…but then I could reveal, well actually, no, it’s a thousand digits of π starting from the two-millionth digit.  And then it’s not random anymore.  So there’s the whole question of what randomness is.  This is one of the central themes that fascinates me: where does this notion come from, where does it lead us in our understanding of the reality we inhabit, and why does it tie in so closely with both the fundamentals of the number system and of particle physics?  And then there’s the difficulty of talking about having two of anything, that in order to have two of anything you have to have a category which those two objects both belong to.  But the categories are always imprecise.  We have to partition spacetime into blocks with ‘fuzzy’ boundaries, and then attempt to match aspects of these blocks up with some ideal which exists in a sort of mental hyperspace, a Platonic realm of sorts.  So we’re projecting these categories onto the universe which actually aren’t intrinsic in the universe; we’re setting out these boundaries, but the boundaries are blurry.  Yet, despite the possible problems this blurriness might cause, on a practical level we’re able to then extract data which fits remarkably well against certain probability distributions. The most ubiquitous and I think the most important one is the Gaussian or bell curve – and this, as we can see from the Erdös-Kac theorem, has a mysterious and fundamental relationship with the number system we’re using to count members of our fuzzy-boundaried categories in the first place. The effectiveness of statistical inference in the hard sciences and the social sciences – I’m sure this would be widely disputed, but I feel there is a mystery there which isn’t really being acknowledged, and it has to do with how we can name and count anything, and how, when we do name, count and measure things they seem to collectively accord with these ideal mathematical blueprints or templates.  That says more about the way our mental hyperspace is being mapped onto the physical universe than about anything intrinsic in the physical universe.

C: When you look at the local you expect to find precision, whereas with the global you’re happy with statistical data.  Here we’re looking at these local, precise conditions and there seems to be randomness ‘built into’ them in a way that’s not immediately comprehensible: After all, they’re not statistical aggregates in any obvious sense.

MW: Yes, the set of positive integers is in a category of its own, there’s just one number system. Yet, it’s as if this entity – if we take the positive integers, the primes and the zeta function as aspects of a single thing, different aspects of the same entity – rather than being a carved-in-stone, unique thing, is actually just one example of a class of things, and we’re able to apply statistical analysis because of that. This is why, when I started finding out about these things, I felt my ‘prime evolution’ thing might have something in it, this idea of the number system being a frozen state of something which had previously inhabited many different states.  I’ve had certain quite critical, serious-minded people react to some of my more sensational suggestions by saying, well all this number theory and physics, there’s nothing mysterious at all — the universe follows mathematical laws, so of course we’d expect certain aspects of number theory to show up in the physical world.  If they’d look a bit deeper into this, they’d see what I meant:  yes, it’s not surprising, given that maths underlies all of physics, that we might get, say, particular values of the zeta function showing up in string theory, or the theory of integer partitions relating to Bose-Einstein condensates or whatever: you get these odd little instances of number theory/physics correspondence; I’ve catalogued a lot of these in my web-archive.  But that’s not the really interesting stuff.  What’s much more surprising is the way physics seems to be pointing the way for understanding the zeta function, and often this is statistical or stochastic physics, as if the zeta function – and in some sense, then, the number system — is just one example of a more general phenomenon.  And I don’t think anyone disputes the spectral nature of the Riemann zeros now.  But it’s not one archetypal ubiquitous spectrum we see showing up all over physics.  If we saw ‘the zeta spectrum’ – as it might be called – everywhere, then it would somehow feel a lot less mysterious.  We’d probably feel quite comfortable with such an affirmation of the old idea that the number system directly underlies the structure of the physical universe.  But the Riemann zeros take the form of an almost disconcertingly arbitrary-looking spectrum, never known of by humans prior to the late 1850’s.  In the very recent past we’ve been confronted with the fact that it has all the fingerprints of membership in certain classes, very wide classes, of very specific physical systems, as if it’s just one element of a whole class, a population of things.  So it’s a bit like the way you might be able to, based on the postcode of a UK resident, predict certain things about his or her attitudes, abilities, tastes, whatever – because you’ve got statistical information about the population, you can make plausible hypotheses about this specific individual.

And it’s as if the primes-zeta entity, whatever you want to call it, despite its seemingly fundamental, unique status, is just one individual in a wider class of things.  But the space in which that class exists is something we haven’t even begun to imagine might exist, or we haven’t got any access to.  So we have this image of a frozen system, something congealing into a state, and then…it’s as if you walked into a concert hall and caught the last note of a symphony, and everybody’s applauding ecstatically and you’re wondering, what’s all the fuss about?  You didn’t witness the process that led up to that last note, and it’s like, with the prime numbers, we’re just walking in on the last moment, the culmination of something.  As if there was a  whole ‘symphony’ that led up to that, and humanity may be on the verge of revealing it. * C: All of the foregoing seems to suggest that what we think of as simple and elegant foundations may in fact be the eventual product of something which is rather complex, even beyond our comprehension.  So we’d have to separate out what seems simple and elegant to us, from what is actually fundamental in the universe, and this is another sense in which mathematics mirrors the condition of theoretical physics, in which, characteristically, the further we go towards the fundamental, the stranger things become (string theory being a case in point).

Rather than defining the primes on the basis of the supposedly fundamental and simple number line, in fact it seems that, when we look through this complex theoretical-mathematical prism you have described, there’s actually something more fundamental about the primes. The primes themselves produce...

MW: ...the number line, yes, you can see it that way.  I came up with this naïve idea, before I really learned any of the more serious stuff, this was after I had been thinking about the Erdös-Kac theorem, the primes and the Gaussian distribution, but before I ‘experienced’ the dynamical aspect of the primes. I was thinking about how we tend to construct the primes.  We’re taught to construct the number line starting with one and then using the Peano axioms, you know, there’s an axiom that basically says, what ever  number you arrive at you can always add another one to it.  And I thought, hold on, where does this come from, this idea that you can always add another one, and I started to question that as something that might not be as obvious as it first seems.  There’s some hidden assumption thereabout order, time or something, I felt.  And I thought, well, there’s an alternate approach we could adopt here, we could start with an infinite alphabet of meaningless symbols, an infinite alphabet of meaningless yet distinct symbols, and then create the dictionary of all possible words of finite length out of that alphabet.

This alphabet of symbols would correspond to the prime numbers.  By combining the symbols in all finite possible combinations, you generate the set of words in your infinitely-long dictionary — this corresponds to the fact that if you combine the primes in all finite multiplicative combinations, you get the set of positive integers.  Except now there’s no sense of order: Because we’re not starting with the positive integers, we don’t need to think of one prime number as being ‘greater than’ another.  The primes are not embedded in the positive integers yet, they’re just these free-floating abstract symbols.  So I used to try and conjure up this image of bubbles floating in an imaginary space, each with an exotic glyph, a symbol from our ‘alphabet of primes’ on it.  The idea is that you can then join any number of these bubbles in any combination, including repeats.  All possible such bubble-clusters are to be found floating somewhere in this space.  Some are larger than others in the sense that there are more bubbles in the cluster — that is, more prime factors — but there’s no sense of a cluster coming ‘before’ or ‘after’ another cluster. It’s only when you cross the Rubicon of deciding which alphabetic symbol is going to be your ‘2’ that you start to create some sense of order. So I had these hints and intuitions — I couldn’t really pin them down to anything very rigorous — that we’ve been thinking about randomness and the fundamentals of reality in a back-to-front way.  We’ve got ourselves into a kind of confusion where everything seems immensely complicated when we delve down to the fundamentals of either the number system — which seems at least partly to inhabit  the realm of psyche — or of the physical world, the world of matter — just open a textbook on analytic number theory or quantum mechanics and you’ll see what I mean.  I felt this issue could be addressed if we examined some of our ‘obvious’ assumptions.  We think we’ve taken the obvious construction — that is, you start with one, then you add one, and then you add another one, this idea you can always add another one.  Rather, what if we start with the primes, and build the number system up that way?  The whole ‘order’ thing then becomes more of a ‘phenomenon’ than something axiomatic...

C: Coincidentally, the ‘legendary’ Dr. Daniel Barker  also devised a notation system for the positive integers based upon prime factorisation, which is very close to what you’re talking about here.7 You have these inseparable lexicographical units from which numbers are composed, and they could be in any order.  He was interested in place value as a culturally-repressive numerical practice, and this was a way of doing away with place value completely.  Each number would just be like a collection of boulders or something.

MW: The lexicographical approach, yes. I’ve tried to get this across to some lay-people I’ve talked to.  There’s the fundamental theorem of arithmetic — literally the most important thing we know about the number system. And no more than 0.1% the population have even heard of it, I’d guess.  It basically says that every integer breaks down uniquely into prime factors.  And we’ve got this strange situation where almost nobody knows this, this simple fact, the most important thing we know about the number system.  This is straying into other territory, but to me, humanity’s relationship with number is rather unhealthy, because we’ve built this entire civilisation around the mathematical sciences, and yet the ordinary population knows nothing of the basics, and often finds mathematics a source of fear and unpleasantness.  I try and conjure up this image of these bubbles, the fact that the clusters can be as large as you want, you can have huge ‘planets’ of prime factor bubbles joined together — there’s no upper size limit. And so something like the greatest common divisor can then be explained very simply, it’s just the intersection, literally where the two clusters intersect.   The least common multiple can be similarly explained.  Prime numbers distinguish themselves from non-prime integers because they are individual bubbles. The integer 1 is the absence of any bubble, the empty background space, the blank page in the “dictionary” I mentioned earlier…  And then you imagine stringing the entire set of clusters out in a line according to this ‘order’ thing, and you start to see that there’s a counterintuitive variation in the sequence — you get small clusters, huge clusters and single bubbles all intermingling according to no sensible scheme. And this is the sort of thing that I’d eventually like to push further out into the public domain just to see what sort of effect it would have, when people start looking at their supposedly familiar number system in this new light.  Because people tend to think of the number system like a row of boxes of cereal in a supermarket, just identical units stacked together, a sort of homogeneous featureless thing that just goes on: each number is just the previous one plus one, there’s nothing much there, nothing of interest.  And it was Frank Sommen, a really remarkable, imaginative Flemish mathematician who I worked with during my PhD studies, who once said to me, every positive integer is a different animal.  I came to see exactly what he meant: each one’s got its own ‘anatomy’, every one’s a different story, and that starts to become apparent as soon as you realise that each integer factors in a unique way into prime numbers. 

PC: You have made numerous interesting observations in the above section of the discussion.

However I will confine myself here to just three important issues.

You mention Timothy Gowers and his contention that the primes somehow feel like experimental data, but at the same time he’s well aware that they are rigidly determined.

This simply reflects the standard reduced view where the primes are interpreted in an absolute quantitative manner.

However appropriately understood, the primes have complementary quantitative and qualitative aspects which dynamically interact in a two-way fashion.

This implies that an important uncertainty principle applies to the very nature of each prime.

So, for example as we have seen with the number 2, its two units possess a relative independent identity (as separate units) in quantitative terms and also a relatively interdependent identity (as shared units) in a qualitative manner.

Therefore from a strict experiential perspective, which is the only way we can truly understand number, the nature of a prime cannot be absolute.

We can however approach close to this absolute state by emphasising the independent quantitative nature of the prime to the almost total exclusion of its qualitative counterpart. And of course this is what happens in the conventional mathematical interpretation of prime numbers.

However as you have illustrated very well we can never attach absolute status to any category of understanding.

One might initially counter that whereas this might strictly apply with respect to physical phenomena, where some doubt as to appropriate classification may always exist, it does not apply to the mathematical world of number.

However this is not in fact the case. The wide held belief in the existing nature of number in itself reflects a tight social consensus of professional mathematicians all abiding by the same rules and procedures.

However this consensus is by no means absolute, especially when someone dares to seriously question these arbitrary rules. For example my notion of a prime differs radically from the present consensus view. So even though we are dealing here with the mathematical category of a prime number, consensus as to its nature is clearly relative.

Just as in experience, one can approach the absolute extreme where a prime appears totally fixed in quantitative terms as rational form, one can also approach the relative extreme in psychological terms, where the same prime now appears fully dynamic as a purely intuitive energy state.

And actual experience necessarily entails a trade-off as between these two extremes, which can change, implying continual transformation with respect to number.

So, just as an uncertainty principle applies to particles in quantum mechanics, it equally applies to number in the relationship between its quantitative and qualitative aspects. And in conventional mathematical terms, the quantitative has blocked out to a remarkable extent any explicit recognition of the holistic qualitative aspect of the primes.

However just as the uncertainty principle applies to the internal nature of each prime, equally it applies externally to the collective nature of all primes.

So we can indeed start by attempting to view the collection of primes in a merely analytic quantitative manner (as absolute unchanging forms). However this is strictly meaningless in the absence of a corresponding qualitative shared relationship existing between such primes.  And this qualitative relationship is provided by the zeta zeros which are then in relative terms understood in a holistic intuitive manner.

So once again in understanding the primes we have the rational extreme approaching completely fixed forms (in quantitative terms) and the corresponding intuitive extreme, where through the zeta zeros they approach a purely elusive formless state (in a qualitative manner).

And a truly comprehensive understanding is subtler still when one allows for the dynamic shifting of reference frames in experience as between quantitative and qualitative (and qualitative and quantitative) respectively.

So equally in internal terms with respect to each individual prime, we can view its component units in an analytic quantitative manner, whereby in complementary terms the prime itself is now viewed relatively in a holistic qualitative fashion; likewise externally with respect to the collection of primes we can view the zeta zeros in an analytic quantitative manner whereby relatively, the primes are now viewed as constituting the factors of the natural numbers in a holistic (qualitative) fashion.

So both in internal and external terms an uncertainty principle exists as to the precise nature of the primes, which can vary as between two extreme points, which are analytic and holistic (and holistic and analytic) with respect to each other.

Indeed this can be used to perhaps clarify somewhat the insight that both you and Lapidus share regarding the “dynamic flow” of the primes.

As we have seen above, the true experience of the primes (externally considered) is inherently dynamic, varying as between the two extremes of absolute rigid forms and purely relative intuitive energy states (represented by the zeta zeros). 

And because of the holistic complementarity of the physical and psychological aspects of reality, this equally entails variation as between the two corresponding extremes of physical forms and energy states respectively.

Thus in both physical and psychological terms, a continual dynamic flow properly constitutes the true nature of the primes. And as number (with respect to its quantitative and qualitative aspects) represents the deepest encoding of reality, this flow is inseparable from all the phenomenal events of everyday life.

Thus the tension between addition and multiplication, which equally reflects the tension as between the quantitative and qualitative aspects of mathematical understanding, cannot be properly resolved until one is able to hold together in a balanced manner in experience these two extremes of the primes, as rigid forms and pure energy states respectively.

Of course the deeper implication here is that a comprehensive mathematical understanding must thereby incorporate the marriage of specialised rational interpretation (related directly to analytic type appreciation) with a highly refined contemplative vision (related directly to pure holistic awareness).

And once again I can only repeat with every fibre of my being that an enormous shadow hangs over the mathematical profession in its total inability to recognise the crucial importance of the holistic aspect, which intimately affects appreciation of every mathematical notion.

In a misguided attempt to create a merely quantitative interpretation of mathematical symbols, it has constructed a hermetically sealed chamber with respect to accepted assumptions and practices effectively insulating itself from any outside criticism. And this represents an extremely unhealthy state for the future development of mathematics!

If I was in dialogue with Michel Lapidus or indeed any leading members of the profession, I would attempt to directly communicate the great urgency of this matter. So they now need to seriously address the very assumptions regarding mathematics that they have blindly accepted heretofore without question.

Put bluntly, these assumptions are no longer fit for purpose. Indeed despite the undoubted marvellous achievements of the quantitative approach, strictly they never were fit for purpose! Unfortunately however, the mathematical community is still a very long way indeed from recognition of this fundamental fact.

So for all their undoubted brilliance and technical expertise, without admitting the equally important role of the holistic aspect in their interpretation, mathematicians cannot hope to grasp, among other things, the true significance of the Riemann Hypothesis.

You mention the Erdös-Kac Theorem in several places, which clearly has had a major influence on your thinking.

I would have some reservations as to the manner in which you attempt to distinguish purely mathematical Gaussian distributions from their physical counterparts, when in the end both are of a strictly relative nature.

However I take your point that because the supply of mathematical data is infinite – though I would prefer to say finitely unlimited – that a much more accurate bell curve can theoretically be obtained to model the distribution of distinct prime factors. But again, this accuracy will always remain of a relative rather than absolute nature. In any case we can never have fully independent data in mathematical terms, which view simply reflects the standard reduced quantitative approach to number.

Once again if data were indeed independent in absolute terms then it would not be possible to relate them in any meaningful sense!

And from a practical perspective it should be stated that the Gaussian distribution is of limited use in making predictions, due to the fact that within the typical range of computation, a number on average contains very few (distinct) prime factors.

So for example with respect to (distinct) prime factors, the mean average for a number composed of 10,000 digits would be about 10. So if we attempted to represent the distribution of these prime factors with a normal bell curve, it would be very far from perfect.

Indeed we would have an inevitable lack of symmetry in terms of the two sections of the curve, left and right of the mean. This is due to the fact that whereas there is an obvious limit to how few (distinct) prime factors a number can contain (i.e. 1, which for a number with 10,000 digits is just 9 less than the mean), effectively there is no upper limit to the maximum number of factors.

In this case, the normal curve would be best for predicting the probability of a number containing those (distinct) prime factors that are centred close to the mean and thereby occur most frequently. So for example there is about a 68% chance that a number chosen at random would contain on average between 7 and 13 (distinct) prime factors. However it would be especially inaccurate in predicting – say – the percentage of numbers containing just one (distinct) prime factor.

However given these qualifications, the Erdös-Kac theorem is remarkable in the way that it can theoretically demonstrate the random nature of the distribution of prime factors.

But from another perspective, it is not really surprising. So for example, in a simple coin tossing experiment where the outcome for each individual (unbiased) coin is random, one would expect with numerous sample trials (of the same size) the mean frequency of H’s to approximate a perfect normal distribution as the number of tosses in each trial continually increases.

So accepting that the occurrence of each individual prime is random, we would be led to expect a similar outcome when calculating the frequency of (distinct) prime factors.

However, the tendency for both distributions (coins and prime factors respectively) to be approximated by a perfect bell curve would not necessarily imply that the two distributions are random in exactly the same manner. So, because it inherently relates to the dynamic nature of number, once again randomness is a concept that lends itself to a relative rather than absolute definition.

From my perspective, what is really interesting however is that the Erdös-Kac Theorem indirectly highlights the fact that there are indeed two complementary ways, in which the relationship between the prime and natural numbers can be understood.

So from the more common external perspective, we can look at the distribution of the individual primes among the natural numbers.

However from the related internal perspective, we can look at the collective distribution of (distinct) prime factors among each natural number.

These two distributions are related to each other in a dynamic complementary manner. And this then further implies that the relationship as between the primes and natural numbers is one of mutual interdependence.

Thus externally, from the quantitative (analytic) perspective, primes appear to determine the magnitude of the natural numbers; however relatively, from the holistic qualitative perspective, it is the reverse with the spacing between the primes apparently determined by the natural numbers.

Then internally, each individual prime appears in qualitative (holistic) terms to be determined by an ordered set of natural numbers; from the quantitative (analytic) perspective, each natural number appears to be determined by a random set of prime factors!

It should be equally possible to provide a corresponding external Erdös-Kac theorem to demonstrate the random nature of the individual primes among the natural numbers.

Thus when n is very large, we could take successive similar sized random samples, preferably in the vicinity of n (that are large in absolute terms though very small relative to n).

The actual number of primes occurring in each sample will then cluster around an average mean value with deviations positive and negative occurring with respect to this value.

And the contention is that the deviations from the mean value are normally distributed, approaching a perfect bell curve when both population and sample size increase without limit.

So these complementary distributions can then be used to demonstrate again the two-way dynamic relationship connecting the complementary notions of randomness and order with respect to the number system.

Thus in terms of the external relationship, though each individual prime is random the collective relationship of these primes, with respect to the number system, is of a highly ordered nature.

Then in terms of the internal relationship, though each individual prime is of a highly ordered nature (so for example that 3 entails 1^st, 2^nd and 3^rd members) the collective relationship of prime factors (with respect to each individual natural number) is of a random nature.

When one understands their true dynamic nature, with an uncertainty principle applying that mirrors closely that of quantum mechanical particles, then there should be little difficulty in appreciating why probabilistic number theory has proven so successful in terms of the primes.

In fact many surprising connections involving probabilistic notions can be made directly with the Riemann zeta function.

For example, the probability that a number chosen at random will be composed of non-recurring prime factors = 1/ζ(2) = 6/π² = .6079 (approx).

This could also be expressed as the probability that a number is square-free or alternatively the probability that a number contains a prime factor (or factors) that occur(s) at most once.  

Then in more general terms, the probability that 1 or more prime factors will occur at most n times = 1/ζ(n + 1) – 1/ζ(n).

Therefore for example the probability that 1 or more prime factors will occur at most twice = 1/ζ(3) – 1/ζ(2) =  .8319… –  .6079… = .224. And in fact there are 23 such numbers from 1 – 100.

Equally 1/ζ(2) is the probability that two whole numbers chosen at random will not contain a common (proper) factor.

And then in more general terms 1/ζ(n) is the probability that n whole numbers chosen at random will not contain a common factor.

So for example, the probability that 3 whole numbers chosen at random will not contain a common factor = 1/ζ(3) = .8319….

You make the very interesting observation here and in the trilogy that the structure relating to the spectrum of zeta zeros looks like one particular example of a class of physical systems, whereas normally we would expect it to be the other way round, with a physical system’s behaviour corresponding to a particular class of mathematical structures.

A considerable problem here relates to the fact that we have been conditioned to consider mathematical structures in fixed absolute terms.

However properly understood as we have seen, number is dynamic (with complementary physical and psychological aspects). And this is due to the fact that number represents the inherent encoding of all phenomena as to their fundamental nature.

So, I have no difficulty therefore, in accepting that the spectrum of zeta zeros represents a specific example of an inherently dynamic mathematical system.

I would then see from this perspective that the physical quantum system (of energy states), which has been the subject of so much Riemannian research, closely resembles the original dynamic mathematical system of zeta zeros. However, at the extreme point where a physical system precisely matches its mathematical counterpart, by definition, it loses any distinct physical identity!

Holistic understanding can then suggest the reason why there is such a close approximation here as between both systems.

As the sub-atomic level lies close to the formless holistic ground of matter, the nature of both physical and mathematical reality becomes much more closely intertwined at this stage. So when one probes deeply within the most transient manifestations of matter, one is left with a physical understanding of reality ever closer to the mathematical (as the encoding of its underlying nature).

And though I would be heavily critical of its mere analytic mode of interpretation, superstring theory testifies very well to this fact, where it constitutes more a strictly mathematical than physical appreciation of nature.

I will now address the issue as to why the spectrum of zeta zeros is apparently associated with a particular case (quantum chaotic) of a more general class of physical systems.

In dynamic terms, number as we have seen, combines both analytic and holistic aspects. Whereas the analytic can be directly identified in physical terms with the classical approach, the holistic aspect by contrast can be more naturally identified with the wave nature of quantum reality.

Thus to properly mirror both dynamic aspects of number, it is reasonable to suppose that it should thereby combine a mixture of the classical and quantum approaches in physical terms. Also, whereas the phenomenal nature of physical reality loses any strict identity below the Planck length, it can still be given a meaning in classical mathematical terms. So quantum chaology, as proposed by Michael Berry, seems to me as an attempt to somehow marry the physical with the mathematical nature of reality at its closest approach to a purely formless state.

Thus I am not surprised that a close correspondence has been shown as between the zeta zeros and the energy levels of a postulated quantum chaotic physical system. However any physical system that exactly replicates the zeros would by definition have a purely mathematical identity (as the hidden encoded nature of the system).

So in this sense, one could truthfully say that the postulated physical system, for which Berry and his colleagues are searching, is in fact none other than the – holistically understood – set of zeta zeros, where the highly dynamic physical nature of reality approaches identity with its mathematical encoding!         

This also implies that the ultimate physical explanation of reality must be therefore of a mathematical nature (with respect to both its analytic and holistic aspects).

And this explanation relates directly as we have seen to the dual dynamic nature of the number system which inherently encodes the nature of reality in quantitative and qualitative terms.

In this respect, current physical theories such as superstring theory are strictly of a secondary nature. And the great limitation of these theories is that, as presently constituted, they are interpreted in a merely analytic manner. So no means presently exist therefore for showing how their findings can properly resonate with a truly integrated appreciation of the overall nature of reality.

It is vital also at this point to recognise a key distinction as between scientific and artistic type appreciation of phenomena.

If we take for example flowers such as roses to illustrate, from the scientific perspective the important requirement is that each individual rose can be identified as a common member of the overall class of roses. In this context, the roses are given a merely impersonal cognitive identity in quantitative terms, thereby enabling them to be employed as scientific data.

However it is somewhat the opposite from the artistic (aesthetic) perspective, where each rose is now recognised in terms of its unique individual qualities which are conveyed through the affectively inspired senses.  

Recent research has concentrated almost exclusively on attempting to demonstrate the physical links with the zeta zeros.

So again whereas this scientific interpretation conforms directly to the analytic aspect of understanding (where the parts are collectively related to the whole) artistic appreciation in complementary terms conforms to the holistic aspect (where the whole is contained in each individual part member).

Therefore when it comes to the holistic interpretation of the zeta zeros (as complementary to the analytic primes) we would expect them to conform to a special case of a more general phenomenon.

However again the deeper message here is that mathematics itself in its understanding of number must now move directly to embrace the qualitative aesthetic aspect of reality, just as the analytic had formerly been identified with the quantitative rational aspect. 

However this realisation then greatly increases the mystery of how the qualitative aspect of phenomena (with unique diversity for each member of a class), which is inherent in all number appreciation, can then be made consistent with the quantitative aspect of interpretation (where all are collectively understood as common members of the same class).

This mystery is then tied up, I believe, with the general class of L-functions, of which the Riemann zeta function serves as the archetypal key member.

It is as if the Riemann zeta function provides us with the basic colours or tones to enable holistic (qualitative) type appreciation of number in a manner that harmonises it – through the requirement that all the zeta zeros lie on the critical imaginary line – with corresponding analytic (quantitative) interpretation.

The other L-functions, that are potentially infinite in number, relate to various ways in which these colours and tones can be configured. And all these in turn have their distinctive zeros and Riemann Hypotheses thereby enabling both holistic and analytic type appreciation to be reconciled with each other in every single case.

And then by extension all these mathematical relationships i.e. L-functions – with respect to their complementary holistic and analytic meanings – can now be profoundly understood as underlining the very means by which artistic sensibility and scientific understanding can in turn be fully harmonised with each other in human experience.

Precious little attention however has been given in research to the psychological role of the zeros, which is of equal importance to the physical aspect.

So this is an issue to which I have devoted considerable attention in recent years.

To your credit, you go much further than your experimental colleagues in recognising that the zeta zeros should represent some deep archetypal imprint of consciousness.

Though we have already addressed this to some degree earlier in the discussion, because of its great importance it may be valuable to address it again.

We recognise the significance of the prime numbers in analytic terms as they form the basis of our quantitative appreciation of reality.

The way to understand the fundamental role of the zeta zeros is to now look at their significance as directly complementary, which befits a dynamic appreciation. 

Thus the great importance of the zeros from a holistic perspective, simply stated, is that they form the basis for our qualitative appreciation of reality.

I should perhaps clarify the nature of this qualitative appreciation again in that it is directly related to the manner we instinctively relate to the world through sensation and feeling!

In earliest childhood, as we have seen, experience is necessarily primitive in nature, where the (unconscious) holistic desire for meaning is directly confused with specific (conscious) phenomena.

So the holistic appreciation of the zeta zeros can best be viewed as achieving the full developmental solution with respect to eradicating the involuntary nature of primitive instinctive desire. Thus through the zeta zeros, the sense recognition of phenomena can be properly combined in a sustained manner with the most refined form of contemplative type awareness. Thus as well as possessing a unique individual identity, each sense phenomenon now continually radiates a numinous quality as an archetype of an eternally present spiritual moment.

At our present stage of evolution, few realistically have the capacity to go beyond mastery of the early zeta zeros. So the understanding in holistic terms of each new zero (on the imaginary scale) can then be seen to represent a further degree of intuitive refinement in the contemplative manner one thereby experiences sense phenomena.

So in the future, I have little doubt that the holistic appreciation of the zeta zeros will come to play a central role in the advanced meditative practice of spiritual programmes of development.

However from another perspective, where reference frames are reversed we have the complementary relationship of the zeta zeros (as analytically understood) and the holistic appreciation of the prime numbers.

Now there is little need to say much here regarding the analytic interpretation of the zeros, which have received intensive investigation in both mathematical and physical terms. So mathematicians are well aware at this level of the important role which the zeros play in relation to the quantitative distribution of the primes.

However the corresponding holistic interpretation of the primes, though of immense potential significance has received little or no attention.

I will try and briefly comment here on its vital importance.

Indeed initially it came as a great surprise, when I realised the enormous significance attached to the holistic notion of the primes and natural numbers (which in dynamic terms are always necessarily intertwined with each other).

It is easy to see in quantitative terms how 1 represents a specific member of the natural number class.

Now the significance of 1 in a holistic context relates to the fact that what is conventionally recognised as mathematics represents just one possible form of interpretation, which is 1-dimensional in nature.

So just as in quantitative terms, 1 represents a specific member of the natural number class (now defined in a qualitative holistic manner).

This entails that there are an unlimited number of other possible interpretations (besides 1) for defining mathematical reality.

So just reflect on this one point for a moment! Though potentially an unlimited set of interpretations exists, what is presently accepted as mathematics is rigidly confined to just one such interpretation (i.e. 1-dimensional).

This interpretation is however unique in that it is the only one that is absolute in nature (where the qualitative aspect is formally reduced in quantitative terms).

All other dimensional interpretations (≠ 1) are of a dynamic relative nature, entailing the interaction of both analytic (quantitative) and holistic (qualitative) aspects.

Then, as I have already stated, appreciation of the simplest of these relative systems (2-dimensional) serves as a prototype for all other possible interpretations.

Therefore, though complexity increases with the higher numbered dimensional interpretations, there is no change in the fundamental rationale of relative dynamic interaction. In general terms, these higher dimensions relate to the increasingly intricate ways the two key polarity sets i.e. internal/external and whole/ part can be dynamically configured in a fully balanced fashion with respect to reality.

What is remarkable is that an equally important set of zeros is associated with the holistic nature of the primes which is related to the circular number system (defined in terms of the unit circle in the complex plane).

So we have here the holistic equivalent to obtaining the roots of unity in the complex plane (with however the default root of 1 ignored).

Just as we have a spectrum of zeta zeros representing energy states, equally we have a more obvious spectrum of energy states relating to all the natural numbers (except 1).

And once again, precisely because mathematics, as presently understood, is confined the 1-dimensional approach, it remains completely blind to these alternative dynamic interpretations.

And the truly marvellous fact is that the Riemann zeta function (when correctly interpreted in dynamic interactive terms) provides the key to relating through the entire complex plane (except 1) the manner in which the analytic (quantitative) and holistic (qualitative) aspects of number interact.

And in holistic terms these zeros, which are related to the holistic circular number system (which I generally refer to as the Zeta 2 zeros), play a complementary role to the Zeta 1 non-trivial zeros that we have already discussed.

So, just as the senses can become increasingly imbued with a refined type of spiritual instinct, likewise reason can equally become imbued with a corresponding type of increasingly refined intuition.

However there is a key difference. Whereas the qualitative appreciation associated with the Zeta 1 zeros operates in an immanent spiritual manner, by which one is better enabled to appreciate the unique individual identity of each phenomenon, the Zeta 2 zeros operate, relatively, in a corresponding transcendent spiritual fashion, by which one is enabled to better appreciate the more general universal features of reality.

However, ultimately in their truest integrated expression, both immanent and transcendent aspects become simultaneously related in a two-way manner with respect to experience.

After concentrating so much effort on getting to the bottom of the Riemann Hypothesis, it then came as something of a shock to realise that this too represented just one specific example of a more general class of mathematical objects i.e. L-functions.

I had discovered the LMFDB (The L-Functions and Modular Forms Database), which comprises a massive highly valuable resource of over 20 million L-functions.

And as modular forms and their accompanying L-functions now seem to be ubiquitous in many areas of mathematics, there is no limit as to the true total existing.

As you know, these L-functions share key features in common with the Riemann Function. They all can be equally expressed as a certain sum over the natural numbers and product over the primes. They all have a symmetry line (through .5 on the real axis) and a functional equation, where values of the function for s > 1 on the right hand axis can be reflected to corresponding values for s < 0 on the left hand axis. And each has its own Riemann Hypothesis with a set of zeros (positive and negative) which are postulated to lie on the imaginary line through .5.

The attempt to prove that the zeros do in fact lie on the line has proven as elusive for all these L-functions (as it has for the Riemann zeta function). And from my perspective, I have little difficulty in appreciating why this is the case, as by the very nature of these functions, a conventional proof (or disproof) is not possible.

In other words in the acceptance of the real number line, the requirement that all the zeros in each case do indeed lie on the imaginary line is already implied by such acceptance.

However the deeper issue for me is the manner in which this cornucopia of other L-functions holistically relates to the Riemann zeta function.

So adding just a little to what I have already said on the matter, my present understanding would mirror that associated with the unlimited set of higher number dimensions for interpreting reality.

Here, 2 served as the prototype for all dynamic relative interpretations of the number system. Thus, though higher dimensional interpretations do entail ever increasing levels of complexity, the basic manner of their overall interpretation remains unchanged.

Therefore in like fashion I would see the Riemann zeta function serving as the prototype for all other L-functions. So the fundamental significance of their holistic zeros for qualitative appreciation would again be similar though allowing for more variation and complexity in experience.

Though an extensive amount of research has taken place in terms of the correspondents to the zeta zeros associated with the Riemann Hypothesis, I am not aware of any similar research with respect to possible physical correspondents of zeros associated with other L-functions. However such correspondents, if identified, are likely to be close relatives of the quantum chaotic system associated with the Riemann zeros.

These L-functions do in fact all represent the ways in which varying arrangements of primes and natural numbers (though potentially of a highly abstruse nature) can be dynamically related through multiplication and addition to each other in a seamless manner.

In fact as I was writing this, I began to see another way – bearing in mind earlier discussion on the notions of randomness and order respectively – of fruitfully highlighting the relationship of the general class of L-functions to their most important original member (i.e. the Riemann zeta function).

Again all L-functions can be defined as both a sum over integers and as a product over primes.

The Riemann zeta function in its sum involves the most ordered arrangement of integers (relating to all the natural numbers) and the most random product arrangement (in the inclusion of all the individual primes).

So, as I have suggested earlier, the Riemann zeta function must be meaningfully interpreted in a relative manner, showing in this case how the extreme notions of order and randomness in terms of the number system are fully related to each other in a dynamic complementary manner.

All other L-functions entail varying configurations with respect to order and randomness. So, on the one hand, the pure order we associate with the succession of natural numbers is sacrificed in varying degrees; likewise the pure random nature that we associate with the full set of individual primes is likewise sacrificed. Thus with respect to the unique configuration of order and randomness involved, each L-function shows how the dynamic relationship as between both aspects is consistently preserved in terms of the number system. 

For example the Dirichlet beta function is one of the best known of the simpler L-functions i.e. Dirichlet L-functions, with its L-series:

1 – 1/3^s + 1/5^s – 1/7^s + 1/9^s – …

Though there is certainly a discernible order here in the sequence of natural numbers, it is not quite as clear as the archetypal pattern associated with the natural numbers.

So, all the even numbers have been discarded and the sign of each successive odd term keeps alternating as between positive and negative respectively.

The corresponding product over primes expression is:

1/(1 + 1/3^s). 1/(1 – 1/5^s). 1/(1 + 1/7^s). 1/(1 + 1/11^s).1/(1 – 1/13^s). …

Here in corresponding fashion we have a departure from the archetypal random pattern of the sequence of individual primes. So the first prime i.e. 2 is omitted and then a definite form of ordering is used to decide the sign of each term. So if on division by 4 a prime leaves a remainder of 3, the term is given a + sign; if however the remainder is 1 (which is the only other option), it is given a – sign. So though the sequence of primes is indeed random, the application of the requisite signs is now based on an ordered procedure.

In this way, with L-functions, we can have endless variation in the manner in which order and randomness are defined (once we depart from the pure archetypal notions of both associated with the Riemann zeta function).

However the crucial point is that for each L-function, a dynamically coherent relationship is established as between a specific type of order and associated type of randomness respectively.

So to sum up, there are an unlimited number of ways in which order and randomness can be dynamically related with respect to the number system (with the Riemann zeta function representing the extreme archetypal example).

This then leads to a key question which Erica Klarreich posed in her excellent “New Scientist” article: “Why do the primes achieve such a delicate balance between randomness and order?”

The holistic explanation for this feature lies in the close relationship as between notions of randomness and order with the corresponding notions of independence and interdependence with respect to the number system.

Therefore from a holistic perspective, it is vital that the relative independence of each individual prime in quantitative terms can be seamlessly synchronised collectively with the corresponding relative interdependence of the primes (with the natural numbers) in a qualitative manner. 

And because number, in terms of both its quantitative and qualitative aspects, represents the inherent encoding of all created phenomena (in physical and psychological terms) this requirement of the number system is likewise necessary so that each phenomenon, while maintaining a unique differentiated identity, can also be seamlessly integrated with all other phenomena in a collective manner.

Again we can see this feature demonstrated especially well at the quantum level of physical reality. Though a random nature attaches to particle interactions in individual terms, a remarkably consistent order characterises their collective behaviour, so that quantum mechanics can make predictions with an extraordinary degree of scientific accuracy. 

And when one can successfully experience reality in this manner (where randomness and order are dynamically integrated), object phenomena then radiate an intense numinous quality. So for example someone advanced in contemplative awareness can appreciate how spirit is immanent in each phenomenon (as its very source), while also recognising that it collectively transcends all material phenomena (as their ultimate goal) in seamless realisation of a shared qualitative interdependence.

This is why I am so confident that eventual discovery of the neglected holistic dimension of mathematics will naturally lead to an authentic spiritual transformation (not directly related to the established religious traditions).

In this context, for one to successfully deal with a highly demanding lifestyle, where spiritual equilibrium can be continually maintained in the midst of intense phenomenal activity, at least some of these other L-functions would be holistically relevant in experience. And once again, this holistic appreciation would tend to naturally occur where understanding of such functions occurs in a dynamic balanced fashion (incorporating both quantitative and qualitative aspects).

And true mastery of the zeros for the Riemann zeta function would greatly facilitate similar mastery for the zeros of related L-functions.